I think .2c - .3c could be reached with manned craft.
New materials are coming online even now to shield the interior of a starship from extremely hard radiation. The future will see even better materials developed. Couple this with magnetic shielding and a whipple type shield traveling ahead of the main craft, and protection improves even more.
Relativistic effects are fairly mild up to around .5c (15.5%) and should be easy to deal with at .2c (2.1%) Navigation could be done with optical computers that could see incoming debris with laser and radar sensors and give warning times of several seconds.
All of these technologies are either here now or are on the near term horizon, I wouldn't care to speculate about technologies 200-300 years distant.